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The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations

Ferruccio ColombiniGuy Métivier — 2008

Annales scientifiques de l'École Normale Supérieure

In this paper we study the Cauchy problem for second order strictly hyperbolic operators of the form L u : = j , k = 0 n y j ( a j , k y k u ) + j = 0 n { b j y j u + y j ( c j u ) } + d u = f , when the coefficients of the principal part are not Lipschitz continuous, but only “Log-Lipschitz” with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant...

A Simple Example of Localized Parametric Resonance for the Wave Equation

Colombini, FerruccioRauch, Jeffrey — 2008

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 35L05, 35P25, 47A40. The problem studied here was suggested to us by V. Petkov. Since the beginning of our careers, we have benefitted from his insights in partial differential equations and mathematical physics. In his writings and many discussions, the conjuction of deep analysis and specially interesting problems has been a source inspiration for us. The research of J. Rauch is partially supported by the U.S. National Science Foundation under...

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